#6575839

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1925 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14,18,23)(15,25,28,20)(16,30,32,21)(17,33,35,22)(19,24,36,41)(26,34,40,43)(27,31,39,44)(29,42,37,38);(6,9)(10,12)(13,15,18,28)(14,20,23,25)(16,31,32,44)(17,34,35,43)(19,37,36,29)(21,39,30,27)(22,40,33,26)(24,42,41,38);(2,4)(13,16,18,32)(14,21,23,30)(15,26,28,40)(17,24,35,41)(19,22,36,33)(20,34,25,43)(27,42,39,38)(29,44,37,31);(5,7,8,11)(6,10,9,12)(16,22)(17,30)(19,36)(21,35)(24,41)(26,44)(27,34)(29,37)(31,40)(32,33)(38,42)(39,43);(13,17,18,35)(14,22,23,33)(15,27,28,39)(16,24,32,41)(19,21,36,30)(20,31,25,44)(26,42,40,38)(29,43,37,34)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483035);